An iron plate is uniformly heated from `20^@C` to `80^@C`. The change in diameter of a circular hole of diameter 1m is `x/100` cm. Find the value of x. `(alpha = 12 xx 10^(-6)//""^@C)`

Explanation

To solve the problem of the change in diameter of a circular hole in an iron plate when heated, we can follow these steps:

### Step 1: Identify the given values
- **Initial temperature (T1)** = 20°C
- **Final temperature (T2)** = 80°C
- **Initial diameter (D)** = 1 m = 100 cm (since we need the answer in cm)
- **Coefficient of linear expansion (α)** = 12 × 10^(-6) °C^(-1)

### Step 2: Calculate the change in temperature (ΔT)
\[ \Delta T = T2 - T1 = 80°C - 20°C = 60°C \]

### Step 3: Use the formula for linear expansion to find the change in diameter (ΔD)
The formula for linear expansion is given by:
\[ \Delta D = D \cdot \alpha \cdot \Delta T \]
Substituting the known values:
\[ \Delta D = 100 \, \text{cm} \cdot (12 \times 10^{-6} \, \text{°C}^{-1}) \cdot (60 \, \text{°C}) \]

### Step 4: Calculate ΔD
\[ \Delta D = 100 \cdot 12 \cdot 60 \cdot 10^{-6} \]
Calculating the multiplication:
\[ \Delta D = 72000 \cdot 10^{-6} \, \text{cm} \]
\[ \Delta D = 0.072 \, \text{cm} \]

### Step 5: Express ΔD in terms of x
The problem states that the change in diameter is given as \( \frac{x}{100} \) cm. We have found that:
\[ \Delta D = 0.072 \, \text{cm} = \frac{72}{100} \, \text{cm} \]
Thus, we can equate:
\[ \frac{x}{100} = \frac{72}{100} \]

### Step 6: Solve for x
From the equation above, we can see that:
\[ x = 72 \]

### Final Answer
The value of \( x \) is **72**.

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